Last updated: 2017-04-01

Trees and tree thinking

  • As soon as you have comparative data, you must consider the relationships among your taxa
    • No longer an option (30+ years of methods)
  • Not concerned in this class with making the tree(s)

You have a tree and data - What do I do now?

Where does your tree come from?

  • Published trees
    • As is
    • Pruned down from published trees
  • Constructed from multiple trees
  • Including tree uncertainty
    • Compare effect of different taxon placements
Trees are hypotheses of relationships. You have hypotheses resting on hypotheses.

Polytomies

Hard:

  • Rapid speciation

Soft:

  • Uncertainty of relationships
  • Adjust the degrees of freedom for tests (Díaz-Uriarte and Garland 1996, 1998)
  • Convert polytomies into very short branches for procedures that can't handle soft polytomies

Polytomies

Is body size associated with home range area in mammals?

Data from Garland et al. (1992)

  • Body mass (km)
  • Home range area (km2)
  • "49lbr" data

Home range vs. body mass

Home range vs. body mass

Home range vs. body mass

Home range vs. body mass

Why phylogenetic comparative methods?

  1. Meet the assumptions of standard statistical tests
    • Any test you can do using "standard" methods has a phylogenetically informed counterpart.
    • What is your evolutionary model?
  2. Ask and answer nuanced evolutionary questions
    • Trait evolution (continuous or categorical)
    • Trait co-evolution
    • Rates of evolution (identify nodes when rates shift significantly)
    • What is your evolutionary model?

Branch lengths \(\approx\) evolutionary model

"Real" branch lengths:

  • Divergence dates (e.g., fossil calibrated)
  • Genetic or character distance (relative)

Arbitrary branch lengths:

  • All = 1
  • Pagel's ultrametric transformation (1992): total height = deepest nesting

Either can be transformed.

Divergence dated

All = 1

Square-root divergence time

Models of evolution

  • Brownian motion
  • Ornstein-Uhlenbeck (OU)
  • Variable rates (ACDC)
  • Pagel's lambda

These models are all accomplished via branch length transformations.

Brownian motion

  • The default, null underlying model for evolution
  • Simplest, most parsimonious model of drift
  • No selection
  • At each step, a trait value is randomly altered based on a normal distribution (mean usually 0) with some variance.

Brownian motion

Simulate 100 Brownian motion random walks with a length of 100 steps.

set.seed(3)
nsim <- 100
t <- 0:100  # time
s2 <- 0.01

X <- foreach(i = 1:nsim, .combine = rbind) %do%
  c(0, cumsum(rnorm(n = length(t) - 1, sd = sqrt(s2))))

Brownian motion

Brownian motion

Brownian motion

Brownian motion

Ornstein-Uhlenbeck model

  • Based on Ornstein & Uhlenbeck (1930)
  • Brownian motion doesn't allow for adaptive evolution (drift only)
  • Simplest model for an evolutionary process with selection (deterministic + stochastic)

The OU model describes the motion of a species in the phenotypic space whereby the species moves randomly within the space, but is influenced by a central tendency such that large deviations from the central optimum receive a stronger force back toward the optimum (Blomberg et al. 2003)

Ornstein-Uhlenbeck model (OU)

\[dX(t) = \alpha[\theta - X(t)]dt + \sigma dB(t)\]

  • Change in trait X at time t is a function of selection \(\alpha[\theta - X(t)]dt\) and drift \(\sigma dB(t)\)
    • \(\alpha\) = strength of selection
    • \(\theta\) = optimum
    • \(\sigma\) = magnitude of drift
  • \(\alpha\) and one or more local optima (\(\theta_1\), \(\theta_2\), etc.) can be estimates
  • Models compared with likelihood ratio tests, AIC, etc.

Ornstein-Uhlenbeck model (OU)

Key readings:

  • Hansen and Martins (1996)
  • Blomberg et al. (2003)
  • Butler and King (2004)

OU evolution along a tree

library(phytools)
set.seed(331)
tree <- pbtree(n = 15, scale = 10)

OU evolution along a tree

OU evolution along a tree

\[dX(t) = \alpha[\theta - X(t)]dt + \sigma dB(t)\]

x <- fastBM(tree,
            a = 0,           # Ancestral state at the root node
            theta = 3,       # Optimum
            alpha = 0.2,     # Strength of selection
            sig2 = 0.1,      # Variance of the BM process
            internal = TRUE) # Return states for internal nodes

OU evolution along a tree

phenogram(tree, x, spread.labels = TRUE, spread.cost = c(1, 0.01),
          main = "scale = 10, alpha = 0.2, sig2 = 0.1")

OU evolution along a tree

OU evolution along a tree

OU evolution along a tree

Estimating an OU model

Rather than simulating data, estimate the values for \(\alpha\) and \(\theta\).

  1. Given a tree and data (body mass in Caribbean anoles)
  2. What model of evolution best fits the data?
    • Brownian motion
    • OU model with a single, clade-wide optimum
    • OU model with three body size optima (S/M/L)
    • OU model with four optima (including an ancestral size)
    • OU model based on multiple colonization events

Estimating an OU model

Evolution toward two local optima:

Estimating an OU model

Estimating an OU model

Estimating an OU model

\(\theta_{small} = 0.25\) mm head length.

ACDC

  • Evolution that either increases or decreases in rate over time.
    • Adaptive radiations followed by phenotypic stasis
    • Proposed by Blomberg et al. (2003)
  • "Early burst" model of Harmon et al. (2010)
  • Didn't really ever catch on as much as OU.

Models of evolution \(\approx\) branch length transformations

Pagel's \(\lambda\): how well does the tree fit the predicted covariance among the trait values

  • \(1\) = Brownian motion
  • \(>1\) = nodes pulled toward the tips (stronger covariance than expected by BM)
  • \(<1\) = nodes pulled toward the root (weaker covariance than expected by BM; more common)

\[\lim_{d\to0} = \mbox{"Star phylogeny"}\]

Pagel's \(\lambda\) branch scaling

Pagel's \(\lambda\) branch scaling

Pagel's \(\lambda\) branch scaling

Pagel's \(\lambda\) branch scaling

Non-R resources

R Resources

Quiz 11-2

Lecture 11-3

References

Blomberg, S. P., T. Garland Jr, and A. R. Ives. 2003. Testing for Phylogenetic Signal in Comparative Data: Behavioral Traits Are More Labile. Evolution 57:717–745.

Butler, M., and A. King. 2004. Phylogenetic Comparative Analysis: A Modeling Approach for Adaptive Evolution. American Naturalist 164:683–695.

Díaz-Uriarte, R., and T. Garland Jr. 1998. Effects of Branch Length Errors on the Performance of Phylogenetically Independent Contrasts. Systematic Biology 47:654–672.

Díaz-Uriarte, R., and T. Garland Jr. 1996. Testing Hypotheses of Correlated Evolution Using Phylogenetically Independent Contrasts: Sensitivity to Deviations from Brownian Motion. Systematic Biology 45:27–47.

Garland, T., Jr, P. H. Harvey, and A. R. Ives. 1992. Procedures for the Analysis of Comparative Data Using Phylogenetically Independent Contrasts. Systematic Biology 41:18–32.

Hansen, T. F., and E. Martins. 1996. Translating Between Microevolutionary Process and Macroevolutionary Patterns: The Correlation Structure of Interspecific Data. Evolution 50:1404–1417.

Harmon, L. J., J. B. Losos, T. J. Davies, R. G. Gillespie, J. L. Gittleman, W. Bryan Jennings, K. H. Kozak, M. A. McPeek, F. Moreno-Roark, T. J. Near, A. Purvis, R. E. Ricklefs, D. Schluter, J. A. Schulte Ii, O. Seehausen, B. L. Sidlauskas, O. Torres-Carvajal, J. T. Weir, and A. Ø. Mooers. 2010. Early Bursts of Body Size and Shape Evolution Are Rare in Comparative Data. Evolution 64:2385–2396.

Uhlenbeck, G. E., and L. S. Ornstein. 1930. On the Theory of the Brownian Motion. Physical Review 36:823–841.